| import Mathlib | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| lemma bounded_geometric_absorption (s : ℕ → ℝ) (M A : ℝ) (hA : 0≤A) | |
| (hM : ∀ n, s n≤M) (hr : ∀ n, s n≤(1/64)*s (n+1)+A*32^n) : | |
| s 0≤2*A := by | |
| have htel : ∀ n, s 0≤(1/64:ℝ)^n*s n+2*A*(1-(1/2:ℝ)^n) := by | |
| intro n | |
| induction n with | |
| | zero => simp | |
| | succ n ih => | |
| have hz := mul_le_mul_of_nonneg_left (hr n) (by positivity : (0:ℝ)≤(1/64:ℝ)^n) | |
| have hp : (1/64:ℝ)^n*32^n=(1/2:ℝ)^n := by rw [←mul_pow]; norm_num | |
| rw [pow_succ, pow_succ] | |
| nlinarith | |
| have hh (n : ℕ) : s 0≤(1/64:ℝ)^n*M+2*A := by | |
| have := mul_le_mul_of_nonneg_left (hM n) (by positivity : (0:ℝ)≤(1/64:ℝ)^n) | |
| have hz : 0≤2*A*(1/2:ℝ)^n := by positivity | |
| nlinarith [htel n] | |
| have hl := (tendsto_pow_atTop_nhds_zero_of_lt_one | |
| (by norm_num : (0:ℝ)≤1/64) (by norm_num : (1/64:ℝ)<1)).mul_const M | |
| have hl := hl.add_const (2*A) | |
| simp only [zero_mul,zero_add] at hl | |
| exact ge_of_tendsto hl (Eventually.of_forall hh) | |
| end | |
| end OAI | |